Does there exist an entire function $F \colon \mathbb{C}\to \mathbb{C}$ such that $F$ is not zero everywhere, $|F(z)|\leq e^{|z|}$ for all $z\in \mathbb{C}$, $|F(iy)|\leq 1$ for all $y\in \mathbb{R}$, and $F$ has infinitely many real roots.

The average age of the 40 members of a computer science camp is 17 years. There are 20 girls, 15 boys, and 5 adults. If the average age of the girls is 15 and the average age of the boys is 16, what is the average age of the adults? $ \text{(A)}\ 26\qquad\text{(B)}\ 27\qquad\text{(C)}\ 28\qquad\text{(D)}\ 29\qquad\text{(E)}\ 30 $

[b]7.1 / 6.3[/b] All integers from 1 to 1966 are written on the board. Allowed is to erase any two numbers by writing their difference instead. Prove that repeating such an operation many times cannot ensure that There are only zeros left on the board. [b]7.2 [/b] Prove that the radius of a circle is equal to the difference between the lengths of two chords, one of which subtends an arc of $1/10$ of a circle, and the other subtends an arc in $3/10$ of a circle. [b]7.3[/b] Prove that for any natural number $n$ the number $ n(2n+1)(3n+1)...(1966n + 1) $ is divisible by every prime number less than $1966$. [b]7.4[/b] What number needs to be put in place * so that the next the problem had a unique solution: [i]“There are n straight lines on the plane, intersecting at * points. Find n.” ?[/i] [b]7.5 / 6.4[/b] Black paint was sprayed onto a white surface. Prove that there are three points of the same color lying on the same line, and so, that one of the points lies in the middle between the other two. [b]7.6 [/b] There are $n$ points on the plane so that any triangle with vertices at these points has an area less than $1$. Prove that all these points can be enclosed in a triangle of area $4$. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3988082_1966_leningrad_math_olympiad]here[/url].

For each $n\in\mathbb{N}$ let $d_n$ denote the gcd of $n$ and $(2019-n)$. Find value of $d_1+d_2+\cdots d_{2018}+d_{2019}$

A closed recticular polygon with 100 sides (may be concave) is given such that it's vertices have integer coordinates, it's sides are parallel to the axis and all it's sides have odd length. Prove that it's area is odd.

Three identical spheres fit into a glass with rectangular sides and bottom and top in the form of regular hexagons such that every sphere touches every side of the glass. The glass has volume $108$ cm$^3$. What is the sidelength of the bottom? [img]https://1.bp.blogspot.com/-hBkYrORoBHk/XzcDt7B83AI/AAAAAAAAMXs/P5PGKTlNA7AvxkxMqG-qxqDVc9v9cU0VACLcBGAsYHQ/s0/2000%2BMohr%2Bp2.png[/img]

A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?

Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. What is the probability that $\lfloor \log_2{x} \rfloor=\lfloor \log_2{y} \rfloor$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to the real number $r$? $\textbf{(A)}\ \frac{1}{8}\qquad\textbf{(B)}\ \frac{1}{6}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2}$

Let $f$ be a quadratic polynomial with integer coefficients. Also $f (k)$ is divisible by $5$ for every integer $k$. Show that coefficients of the polynomial $f$ are all divisible by $5$.

A polynomial with rational coefficients is called [i]integer[/i], if it takes integer values ​​for all integer values ​​of the variable. For an integer polynomial $P$, consider the sequence $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$ a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer $k$ and for all natural $i$, the $i$-th and ($i+2^k$)th members of the sequence are equal). b) Prove that for any periodic sequence consisting of $(- 1)$ and $ 1$ and with a period of some power of two, there exists a integer, polynomial P for which this sequence is $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$

It is given triangle $ABC$. Let internal and external angle bisector of angle $\angle BAC$ intersect line $BC$ in points $D$ and $E$, respectively, and circumcircle of triangle $ADE$ intersects line $AC$ in point $F$. Prove that $FD$ is angle bisector of $\angle BFC$

Find all polynomials of degree $3$ with integer coeffcients such that $f(2014) = 2015, f(2015) = 2016$ and $f(2013) - f(2016)$ is a prime number.

[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions: (i) $\frac{p+1}{2}$ is a prime number. (ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer. [b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.

$ABC$ is triangle. Point $L$ is inside $ABC$ and lies on bisector of $\angle B$. $K$ is on $BL$. $\angle KAB=\angle LCB= \alpha$. Point $P$ inside triangle is such, that $AP=PC$ and $\angle APC=2\angle AKL$. Prove that $\angle KPL=2\alpha$

Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.

In a triangle $ABC$, with $\widehat{A} > 90^\circ$, let $O$ and $H$ denote its circumcenter and orthocenter, respectively. Let $K$ be the reflection of $H$ with respect to $A$. Prove that $K, O$ and $C$ are collinear if and only if $\widehat{A} - \widehat{B} = 90^\circ$.

Let $x,y,z$ be positive numebrs such that $x+y+z\le x^3+y^3+z^3$. Is it true that a) $x^2+y^2+z^2 \le x^3+y^3+z^3$ ? b) $x+y+z\le x^2+y^2+z^2$ ?

Let $p$ be an odd prime. Find all positive integers $n$ for which $\sqrt{n^2-np}$ is a positive integer.

Compute the sum of all real numbers $x$ such that \[2x^6-3x^5+3x^4+x^3-3x^2+3x-1=0.\]

The members of a distinguished committee were choosing a president, and each member gave one vote to one of the $27$ candidates. For each candidate, the exact percentage of votes the candidate got was smaller by at least $1$ than the number of votes for that candidate. What is the smallest possible number of members of the committee?

Let $n$ be an integer, $n \ge 2$. For each number $k = 1, 2, ....., n,$ denote by $a _ k$ the number of multiples of $k$ in the set $\{1, 2,. .., n \}$ and let $x _ k = \frac {1} {1} + \frac {1} {2} + \frac {1} {3} _... + \frac {1} {a _ k}$ . Show that: $$\frac {x _ 1 + x _ 2 + ... + x _ n} {n} \le \frac {1} {1 ^ 2} + \frac {1} {2 ^ 2} + ... + \frac {1} {n ^ 2} $$.

There are $N\ge3$ letters arranged in a circle, and each letter is one of $L$, $T$ and $F$. For a letter, we can do the following operation: if its neighbors are the same, then change it to the same letter too; otherwise, change it so that it is different from both its neighbors. Show that for any initial state, one can perform finitely many operations to achieve a stable state. Here, a stable state means that any operation does not change any of the $N$ letters. (ltf0501)

There are a family of $5$ siblings. They have a pile of at least $2$ candies and are trying to split them up amongst themselves. If the $2$ oldest siblings share the candy equally, they will have $1$ piece of candy left over. If the $3$ oldest siblings share the candy equally, they will also have $1$ piece of candy left over. If all $5$ siblings share the candy equally, they will also have $1$ piece left over. What is the minimum amount of candy required for this to be true?

$ ABC$ is an acute-angled triangle. $ M$ is the midpoint of $ BC$ and $ P$ is the point on $ AM$ such that $ MB \equal{} MP$. $ H$ is the foot of the perpendicular from $ P$ to $ BC$. The lines through $ H$ perpendicular to $ PB$, $ PC$ meet $ AB, AC$ respectively at $ Q, R$. Show that $ BC$ is tangent to the circle through $ Q, H, R$ at $ H$. [i]Original Formulation: [/i] For an acute triangle $ ABC, M$ is the midpoint of the segment $ BC, P$ is a point on the segment $ AM$ such that $ PM \equal{} BM, H$ is the foot of the perpendicular line from $ P$ to $ BC, Q$ is the point of intersection of segment $ AB$ and the line passing through $ H$ that is perpendicular to $ PB,$ and finally, $ R$ is the point of intersection of the segment $ AC$ and the line passing through $ H$ that is perpendicular to $ PC.$ Show that the circumcircle of $ QHR$ is tangent to the side $ BC$ at point $ H.$

Find all positive integers $n \ge 3$ such that there exists an $n$-gon with vertices on lattice points of the coordinate plane and all sides of equal length.